I am studying the $SU(3)$ flavour symmetry and I'm reading that we use the fact that hypercharge and isospin are additive quantum numbers in order to decompose the tensor products of the fundamental representations $3\otimes3\otimes3$ and $3\otimes{\overline3}$ in the direct sum of the irreducible components.

I don't understand why, from a physical and mathematical points of view, hypercharge and isospin should be additive.

  • $\begingroup$ Hypercharge yes as it is an eigenvalue but isospin is like quantum angular momentum and only one of the components is additive. $\endgroup$ – ZeroTheHero Mar 15 '17 at 15:48
  • $\begingroup$ In terms of Gell-Mann matrices, have you inspected the 3 isospin generators and the hypercharge generator? Do they commute? How do they act on the 3 components of a triplet flavor vector? $\endgroup$ – Cosmas Zachos Mar 15 '17 at 20:10

The (strong) hypercharge operator $Y$ and the (strong) isospin operators $(I_1,I_2,I_3)$ are generators of a $u(1)\oplus su(2)$ Lie subalgebra of the Lie algebra $su(3)$ of flavor symmetry. A Lie algebra is a vector space, and hence has a linear structure. Representations can be decomposed in eigenspaces for $Y$ and $I_3$.


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